Quantitative Portfolio Strategy

Quantitative Portfolio Strategy Lev Dynkin 201-524-2839 ldynkin@lehman.com Tony Gould, CFA 212-526-2821 agould@lehman.com CURRENCY HEDGING IN FIXED INCOME PORTFOLIOS Introduction Portfolio managers typically minimize currency exposure in bond portfolios with the use of foreign-exchange forward transactions (forwards). However, the use of forwards cannot entirely eliminate currency volatility, since fluctuations in underlying asset values will lead the portfolio to be either over- or under-hedged. In addition, the use of forwards changes portfolio interest-rate exposures. In this article, we discuss the currency volatility and interest-rate exposures of currency-hedged securities and portfolios and the risk introduced against hedged indices by not replicating index hedges. We also discuss the effect of exchange rate fluctuations on the relative country allocations of portfolios managed against hedged benchmarks. We discuss the methods that investors can use to minimize risk versus hedged benchmarks. Finally, we discuss enhancements that we are making to the analytical information that Lehman Brothers provides for hedged portfolios and indices. The Interest Rate Exposure in Currency Hedges The exchange rate at which a forward is struck (forward rate) is the spot exchange rate adjusted for the interest-rate differential between the two currencies (forward points). This is a necessary no-arbitrage condition, illustrated in Figure 1. In the example in Figure 1, the forward rate is mis-priced, since it does not reflect the interest rate differential between the two currencies, so an investor can buy euros forward too cheaply. As the interest-rate differential between two currencies changes, so does the forward rate, to eliminate arbitrage opportunities. Consider another investor, who holds a 10-year U.S. Treasury yielding 4.2% and hedges the US$ exposure into euros with a one-year forward. As the differential between euro and U.S. interest rates fluctuates, so will the pricing of the forward points. This investor is effectively long one-year euro interest rates and short one-year US$ rates. The currency hedge, therefore, has an effect on the duration exposure of the investor's euro and US$ portfolios. Accordingly, an investor who chooses a different tenor of hedge from a currency-hedged index is assuming active interest-rate risk. This example demonstrates another effect of currency hedging on a portfolio. The yield on the portfolio has also changed. Assuming that the forward rate is now correctly priced Figure 1. Covered Interest Arbitrage Example $/Euro Spot Rate: 1.10 $/Euro 1-Year Forward Rate: 1.08 1 year Deposit Rate - Euro 2.00% - US$ 1.00% Today Borrow 100 Euros for 1 Year at 2% Sell Proceeds for US$ and Invest at 1% Sell $110*(1.01) 1 Year Forward One Year Later Repay Euro Loan Receive Euros from Forward (=110*1.01/1.08) Profit (EUR102) EUR102.87 EUR0.87 Lehman Brothers PLEASE SEE IMPORTANT ANALYST 1 CERTIFICATIONS ON LAST PAGE August 11, 2003

and using the same deposit rates, the yield on the hedged 10-year Treasury is now 5.2%, reflecting the interest-rate differential. More generally, the hedged yield on a security can be expressed approximately as follows: Hedged-yield = Bond yield + (Base currency interest rate - Local interest rate) (1) The duration of a hedged instrument can be expressed approximately as: Hedged-bond duration = Bond duration - tenor of bond hedge (2) The duration decrease in local currency is offset by a corresponding increase in base currency duration. The Exchange Rate Exposure in Currency Hedges An investor typically calculates the amount of local currency to be sold forward to hedge a security's foreign exchange exposure in one of two ways. Either the expected future value of the security at the forward date is used, or the current market value of the security is used. 1 Since the end-of-period market value of the security being hedged is not known, unless this period is equal to the security's maturity, this hedge is not perfect (see exception below). If, for example, a euro-based investor hedges a US$ bond with a market value of $1,000 through the sale of a forward of equal magnitude and the bond rises to $1,100, the investor now has an exposure of $100 to the US$/euro exchange rate. In practice, many investors periodically re-hedge their portfolios to reduce currency exposures, though as we shall see, this may introduce tracking error versus a hedged index. Investors who use longer tenors for currency hedges without periodical re-hedging will tend to have larger currency exposures and currency return volatilities. 2 A foreign security with predictable cash flows may be perfectly currency hedged if the bond is held to maturity and each future cash flow is separately hedged back into base currency. As the example above has shown, since the pricing of the forwards reflects the interest rate differential between the two currencies, this transaction would exchange the interest rate risk of the foreign security for base currency interest rate risk. This would defeat the purpose of owning the foreign security, unless the purpose is to create a new synthetic basecurrency bond, cheaper than a conventional substitute. 3 The size of the currency hedge for Lehman Brothers indices is effectively established at the beginning of each month, when the expected future value of all non-base currency exposures is hedged for one month. The index is not re-hedged until the end of the month and, therefore, to the extent that bond markets move away from their expected 1 Some managers prefer to hedge current rather than future values, entering into currency swaps (a spot purchase and forward sale of currency) when purchasing new securities. 2 For example, a euro-based investor hedging the U.S. component of the Lehman Global Aggregate with a 3-month hedge, with no intra-period adjustment, incurred a currency loss equal to 0.11% for the three months ending June 30, 2003. 3 In practice, the arbitrage advantage of a synthetic bond needs to be significant to offset its substantially reduced liquidity. Lehman Brothers 2 August 11, 2003

month-end values, will be exposed in part to currency movements. 4 The non-deterministic portion of the currency returns for a given country's hedged bond market index is given by: (Local bond market return - Expected bond market return) (FX appreciation) (3) The historical volatility and return of the currency component for a selection of hedged indices are show in Figure 2b. This analysis suggests that currency volatility has a limited effect on the overall volatility of currency-hedged indices; nevertheless, it should not be ignored, especially for single-country bond portfolios hedged into the base currency. Furthermore, there is a marked tendency for spikes in bond market volatility to accompany spikes in currency volatility, increasing the overall volatility of currency returns (this can be seen from Equation 3, in which both bracketed terms would increase). For example, in December 2002 and May 2003, a substantial return on the U.S. Aggregate was accompanied by a large fall in the dollar, leading to an average currency loss for eurohedged investors of 9 bp per month. Figure 2c suggests that investors who wish to minimize overall portfolio risk should not be overly concerned about the volatility of the hedged currency return. Eliminating currency risk entirely can actually increase portfolio risk, as one source of portfolio diversification (at least at these low effective concentrations) is removed. The Timing of Currency Hedges Putting aside those investors who choose to take active currency exposure, investors will incur tracking error versus hedged indices to the extent that hedging methods differ from the treatment of hedges in Lehman Brothers indices. In particular, hedgers may choose 4 For more information on index rules for hedged indices, see "Index Rules for Currency Hedging and Currency Returns," Global Relative Value, June 9, 2003, Lehman Brothers. Figure 2. Volatility of Currency-Hedged Bond Indices Three Years Ending June 30, 2003 Base Annual Annual Currency Return (%) Std Dev (%) 2a. Total Return Global Aggregate USD 7.74 2.74 U.S. Aggregate EUR 10.57 3.23 U.S. Aggregate JPY 6.72 3.28 2b. Currency Return Global Aggregate USD 0.05 0.05 U.S. Aggregate EUR -0.11 0.12 U.S. Aggregate JPY -0.02 0.09 2c. Total Return Less Currency Return Global Aggregate USD 7.69 2.73 U.S. Aggregate EUR 10.69 3.28 U.S. Aggregate JPY 6.74 3.27 Lehman Brothers 3 August 11, 2003

to establish and roll over their hedges at a different time of the month. This may be unavoidable in the middle of the month if the investor establishes a portfolio or receives a large cash flow. Such an investor is now faced with two choices. The most obvious choice is to hedge the current or future expected currency exposure. But the index will itself be partially unhedged, due to bond market fluctuations away from month-end expected values. The currency volatility of the hedged indices given above provides a guide to the potential tracking error such a choice would introduce. Therefore, some investors will prefer the second choice: match the index method and leave part of the currency exposure unhedged. To replicate the index return mid-month, the investor will need to sell the expected forward value of all local currency bonds, computed as at the beginning of the month, for month-end value. 5 Our analytic systems (detailed below) enable users to obtain these values for each bond, as well as aggregate values for each currency. The Tenor of Currency Hedges Some investors choose to use forwards longer (or shorter) than the one-month tenor used by the indices. We have previously examined the implications of selecting alternative hedging tenors in some detail. 6 We noted that implementing a longhorizon hedging strategy represented a view on the direction of relative interest rates between the hedging and base currencies. Investors face a trade-off between increased tracking error and potential increased return. A longer tenor hedging strategy can, depending on the relative shapes of yield curves, increase carry return relative to a shorter tenor. In addition, such a strategy will lead to modestly reduced transaction costs, as forwards are rolled less frequently. An empirical study of various hedging tenors concluded that longer-horizon hedging strategies (up to six months), designed to maximize carry return, have delivered positive information ratios. In this study, we examined the performance of hedging strategies independent from the performance of the underlying bond portfolio. In practice, managers utilizing longer tenor hedges would likely adjust hedges periodically, as movements in bond values create unintended currency exposures. The Effect of Currency Fluctuations on Index Weights All Lehman Brothers indices used in the computation of reported returns (the returns universes) are re-constituted every month according to the market capitalization of the underlying securities that compose the index. For global indices, market values are all converted to the base currency. Accordingly, the changes in index country weights from month to month will depend on, among other factors, monthly fluctuations in exchange rates. Country weights for hedged indices are set to be equal to unhedged indices. Therefore, an investor with a perfect index-replicating multi-currency-hedged portfolio will need to make adjustments to country weights at month-end to preserve the integrity of the replication. For example, following the euro s 5.4% rise against the dollar in May 2003, the euro component of the Global Aggregate rose by 1.5%. An investor would need to make portfolio adjustments to reflect changes in index composition anyway, due to bonds' 5 Ibid. for details of the calculation of the expected forward value. 6 Long-Horizon Currency Hedging in Global Index Portfolios, Global Relative Value, January 7, 2002, Lehman Brothers. Lehman Brothers 4 August 11, 2003

entering and exiting the index, but currency volatility is likely to be the largest contributor to changes in country composition. Figure 3 shows the changes in the currency composition of the Global Aggregate Index for the year to date. Minimizing the Tracking Error Effect of Currency Exposure for Hedged Indices We have shown that investors can replicate the currency exposure of hedged bond indices by matching the tenor and timing of currency hedges. A final potential source of tracking error can arise from executing hedging transactions at prices different from those used for index calculations. While these price differences may be small, their large portfolio weight can cause a meaningful performance shortfall. This is especially true for portfolios composed largely of non-base currency bonds. Investors can now use the Lehman Brothers Perfect Index Execution Service, which guarantees execution of one-month hedges at the exact forward rate used for index calculations. 7 Enhancements to Lehman Analytics for Hedged Portfolios and Indices Lehman Brothers provides transparent index analytics for hedged and unhedged bond indices. We have recently published information on the calculation of hedged returns for currency-hedged indices and added a new field to index analytics that provides the amount of the hypothetical hedges that are implied in hedged indices. 8 We will be adding further functionality to analytic systems, which will provide investors with additional information to assist with the replication of currency-hedged indices. New fields will provide the amount of the index hedge in base and local currency, the amount of the effective currency exposure in the index intra-month for each currency, and the unrealized profit/loss on the hedge. Future enhancements will demonstrate the effect of currency-hedging on portfolio and index yields and duration exposures. Managing and Calculating the Currency Exposure for Hedged Indices The following provides a practical outline of the steps needed to replicate the performance of currency-hedged indices, including descriptions of fields that will be available shortly in index analytics. Figure 4 provides an example of these calculations. 7 See "Index Rules for Currency Hedging and Currency Returns" for further information on the PIE service. 8 Ibid. Figure 3. Currency Composition of the Lehman Brothers Global Aggregate Index in 2003, % 12/31 1/31 2/28 3/31 4/30 5/31 6/30 U.S. Dollar 45.4 44.8 44.7 44.6 44.2 43.1 43.3 Euro 28.4 29.2 29.3 29.4 29.6 31.1 31.0 Japanese Yen 18.5 18.2 18.3 18.3 18.3 17.7 17.7 GB Pound 3.8 3.9 3.7 3.7 3.8 3.8 3.8 Canada Dollar 1.6 1.7 1.7 1.7 1.8 1.9 1.8 Other 2.3 2.3 2.3 2.3 2.3 2.5 2.5 Lehman Brothers 5 August 11, 2003

The index is hedged at 4 pm on the last business day of every month, using a one-month forward. The amount of the hedge is given by RUMVHedgeB. 9 Managers who wish to replicate the index fully will need to replicate the foreign exchange exposure of the index, RUFXExpsr. This arises from the mismatch between the current (local) market value of securities and the expected month-end value (representing the amount of the hedge). RUMVTotLc is the market value of a bond in local currency, including cash generated from that security during the month. RUMVTotLc = (Price+Accrued Interest) * (RUOutLoc/100) + RUMVCashL RUMVTotal is the market value of a local bond hedged into base currency. This comprises the market value of the security (in base currency terms) and the unrealized profit/loss on the forward foreign exchange contract (the sale of local currency back into base). 9 Ibid, for details of the calculation of this value. Figure 4. Currency Hedging Example Bond CUSIP: 69352BAC Index Base Currency: Euros Analytics Field 6/30/03 7/17/03 Price 115.623 114.121 Accrued 4.427 0.396 Par Value RUOutLoc 314,202 295,221 EUR/US$ - Spot Rate 0.870815 0.894174 EUR/US$ - Fwd Rate 0.871569 0.894501 Market Value Bonds - Local Currency 377,200 338,078 Market Value Bonds - Base Currency RUMVSecry 328,471 302,301 Market Value cash - Local Currency RUMVCashL 0 32,968 Market Value cash - Base Currency RUMVCash 0 29,479 Market Value Bonds + Cash - Local* 377,200 371,046 Market Value Bonds + Cash - Base 328,471 331,780 Market Value of Hedge Fwd sale US$ - Local RUMVHedgeB 379,104 379,104 Fwd sale US$ - Base (329,913) (338,886) Fwd Purchase Euros 329,913 329,913 Profit on Hedge (Base) RUHedgePL (8,694) Total Market Value of Hedged Bond Total Value - Local RUMVTotLc 377,200 371,046 Total value - Base RUMVTotal 328,471 323,086 Currency Exposure - Local RUFXExpsr** (1,905) (8.058) * The hedge amount is calculated at the beginning of the month from the expected value of the bond as at month-end. This amount includes any expected security cash flows. ** Since the size of the hedge is based on the month-end expected value of the security, it will be slightly larger than the current market value, creating a small currency exposure. Lehman Brothers 6 August 11, 2003

RUMVTotal = RUMVSecry + RUMVCash + RUHedgePL (see next) RUHedgePL is the unrealized profit/loss on the outstanding forward contract. For the returns universe, the size of the forward (in local currency) is computed at the beginning of each month, shown as RUMVHedgeB. RUHedgePL = RUMVHedgeB * (Fwd RateEnd local/base - Fwd Rate Begin local/base ) Rate local/base = Number of local currency units per unit of base currency. RUFXExpsr is the amount by which a bond is not fully hedged due to fluctuations in its value (including cash) away from the predicted month-end value. This amount can be aggregated to give the total currency exposure for each index currency. RUFXExpsr = RUMVTotLc - RUMVHedgeB Conclusion It is not possible to eliminate absolute currency volatility entirely from foreign bond portfolios. However, this is neither desirable nor necessary. One can imagine an extreme dynamic hedging strategy in which all currency risk would be eliminated as soon as it arose. Every fluctuation of a bond away from its expected value would give rise to a currency transaction to eliminate the resulting currency risk. Such an exercise would be time consuming and costly and, in our analysis, would increase both absolute and relative volatility. Index replication of currency-hedged indices is possible, given our transparent index methodology and analytic systems. In addition, active managers can use currency hedges as an additional alpha-generation tool, since a hedge has both a yield and a duration effect on portfolio exposures. Lehman Brothers 7 August 11, 2003

Lehman Brothers Fixed Income Research analysts produce proprietary research in conjunction with firm trading desks that trade as principal in the instruments mentioned herein, and hence their research is not independent of the proprietary interests of the firm. The firm s interests may conflict with the interests of an investor in those instruments. Lehman Brothers Fixed Income Research analysts receive compensation based in part on the firm s trading and capital markets revenues. Lehman Brothers and any affiliate may have a position in the instruments or the company discussed in this report. The views expressed in this report accurately reflect the personal views of Tony Gould, the primary analyst responsible for this report, about the subject securities or issuers referred to herein, and no part of such analyst s compensation was, is or will be directly or indirectly related to the specific recommendations or views expressed herein. The research analysts responsible for preparing this report receive compensation based upon various factors, including, among other things, the quality of their work, firm revenues, including trading and capital markets revenues, competitive factors and client feedback. Any reports referenced herein published after 14 April 2003 have been certified in accordance with Regulation AC. To obtain copies of these reports and their certifications, please contact Larry Pindyck (lpindyck@lehman.com; 212-526-6268) or Valerie Monchi (vmonchi@lehman.com; 44-(0)207-011-8035). Lehman Brothers usually makes a market in the securities mentioned in this report. These companies are current investment banking clients of Lehman Brothers or companies for which Lehman Brothers would like to perform investment banking services. Publications L. Pindyck, B. Davenport, W. Lee, D. Kramer, R. Madison, A. Acevedo, K. Kim, V. Monchi, C. Rial, K. Banham, G. Garnham, D. Cappadona This material has been prepared and/or issued by Lehman Brothers Inc., member SIPC, and/or one of its affiliates ( Lehman Brothers ) and has been approved by Lehman Brothers International (Europe), regulated by the Financial Services Authority, in connection with its distribution in the European Economic Area. This material is distributed in Japan by Lehman Brothers Japan Inc., and in Hong Kong by Lehman Brothers Asia Limited. This material is distributed in Australia by Lehman Brothers Australia Pty Limited, and in Singapore by Lehman Brothers Inc., Singapore Branch. This material is distributed in Korea by Lehman Brothers International (Europe) Seoul Branch. This document is for information purposes only and it should not be regarded as an offer to sell or as a solicitation of an offer to buy the securities or other instruments mentioned in it. No part of this document may be reproduced in any manner without the written permission of Lehman Brothers. We do not represent that this information, including any third party information, is accurate or complete and it should not be relied upon as such. It is provided with the understanding that Lehman Brothers is not acting in a fiduciary capacity. Opinions expressed herein reflect the opinion of Lehman Brothers and are subject to change without notice. The products mentioned in this document may not be eligible for sale in some states or countries, and they may not be suitable for all types of investors. If an investor has any doubts about product suitability, he should consult his Lehman Brothers representative. The value of and the income produced by products may fluctuate, so that an investor may get back less than he invested. Value and income may be adversely affected by exchange rates, interest rates, or other factors. Past performance is not necessarily indicative of future results. If a product is income producing, part of the capital invested may be used to pay that income. Lehman Brothers may, from time to time, perform investment banking or other services for, or solicit investment banking or other business from any company mentioned in this document. 2003 Lehman Brothers. All rights reserved. Additional information is available on request. Please contact a Lehman Brothers entity in your home jurisdiction.